metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D28.1D4, C23.6D28, M4(2)⋊1D14, Dic14.1D4, C4.78(D4×D7), C4.D4⋊1D7, C28.91(C2×D4), D28⋊4C4⋊1C2, (C2×D4).13D14, C8.D14⋊5C2, C7⋊1(D4.9D4), (C2×C28).3C23, C14.15C22≀C2, D4⋊6D14.2C2, C28.17D4⋊1C2, (C4×Dic7)⋊1C22, C4○D28.1C22, (C22×C14).19D4, C22.10(C2×D28), (D4×C14).13C22, (C7×M4(2))⋊8C22, C2.18(C22⋊D28), (C2×Dic14)⋊12C22, (C7×C4.D4)⋊3C2, (C2×C14).20(C2×D4), (C2×C4).3(C22×D7), SmallGroup(448,280)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D28.1D4
G = < a,b,c,d | a28=b2=1, c4=a14, d2=a21, bab=a-1, cac-1=a15, ad=da, cbc-1=a21b, dbd-1=a7b, dcd-1=a7c3 >
Subgroups: 972 in 152 conjugacy classes, 39 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C42, C22⋊C4, M4(2), SD16, Q16, C2×D4, C2×D4, C2×Q8, C4○D4, Dic7, C28, D14, C2×C14, C2×C14, C4.D4, C4≀C2, C4.4D4, C8.C22, 2+ 1+4, C56, Dic14, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C22×D7, C22×C14, D4.9D4, C56⋊C2, Dic28, C4×Dic7, C23.D7, C7×M4(2), C2×Dic14, C4○D28, D4×D7, D4⋊2D7, C2×C7⋊D4, D4×C14, D28⋊4C4, C7×C4.D4, C8.D14, C28.17D4, D4⋊6D14, D28.1D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C22≀C2, D28, C22×D7, D4.9D4, C2×D28, D4×D7, C22⋊D28, D28.1D4
►On 112 points
Generators in S112
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28)(29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112)
(1 55)(2 54)(3 53)(4 52)(5 51)(6 50)(7 49)(8 48)(9 47)(10 46)(11 45)(12 44)(13 43)(14 42)(15 41)(16 40)(17 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(28 56)(57 110)(58 109)(59 108)(60 107)(61 106)(62 105)(63 104)(64 103)(65 102)(66 101)(67 100)(68 99)(69 98)(70 97)(71 96)(72 95)(73 94)(74 93)(75 92)(76 91)(77 90)(78 89)(79 88)(80 87)(81 86)(82 85)(83 112)(84 111)
(1 35 8 56 15 49 22 42)(2 50 9 43 16 36 23 29)(3 37 10 30 17 51 24 44)(4 52 11 45 18 38 25 31)(5 39 12 32 19 53 26 46)(6 54 13 47 20 40 27 33)(7 41 14 34 21 55 28 48)(57 104 64 97 71 90 78 111)(58 91 65 112 72 105 79 98)(59 106 66 99 73 92 80 85)(60 93 67 86 74 107 81 100)(61 108 68 101 75 94 82 87)(62 95 69 88 76 109 83 102)(63 110 70 103 77 96 84 89)
(1 104 22 97 15 90 8 111)(2 105 23 98 16 91 9 112)(3 106 24 99 17 92 10 85)(4 107 25 100 18 93 11 86)(5 108 26 101 19 94 12 87)(6 109 27 102 20 95 13 88)(7 110 28 103 21 96 14 89)(29 72 50 65 43 58 36 79)(30 73 51 66 44 59 37 80)(31 74 52 67 45 60 38 81)(32 75 53 68 46 61 39 82)(33 76 54 69 47 62 40 83)(34 77 55 70 48 63 41 84)(35 78 56 71 49 64 42 57)
G:=sub<Sym(112)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,55)(2,54)(3,53)(4,52)(5,51)(6,50)(7,49)(8,48)(9,47)(10,46)(11,45)(12,44)(13,43)(14,42)(15,41)(16,40)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(28,56)(57,110)(58,109)(59,108)(60,107)(61,106)(62,105)(63,104)(64,103)(65,102)(66,101)(67,100)(68,99)(69,98)(70,97)(71,96)(72,95)(73,94)(74,93)(75,92)(76,91)(77,90)(78,89)(79,88)(80,87)(81,86)(82,85)(83,112)(84,111), (1,35,8,56,15,49,22,42)(2,50,9,43,16,36,23,29)(3,37,10,30,17,51,24,44)(4,52,11,45,18,38,25,31)(5,39,12,32,19,53,26,46)(6,54,13,47,20,40,27,33)(7,41,14,34,21,55,28,48)(57,104,64,97,71,90,78,111)(58,91,65,112,72,105,79,98)(59,106,66,99,73,92,80,85)(60,93,67,86,74,107,81,100)(61,108,68,101,75,94,82,87)(62,95,69,88,76,109,83,102)(63,110,70,103,77,96,84,89), (1,104,22,97,15,90,8,111)(2,105,23,98,16,91,9,112)(3,106,24,99,17,92,10,85)(4,107,25,100,18,93,11,86)(5,108,26,101,19,94,12,87)(6,109,27,102,20,95,13,88)(7,110,28,103,21,96,14,89)(29,72,50,65,43,58,36,79)(30,73,51,66,44,59,37,80)(31,74,52,67,45,60,38,81)(32,75,53,68,46,61,39,82)(33,76,54,69,47,62,40,83)(34,77,55,70,48,63,41,84)(35,78,56,71,49,64,42,57)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112), (1,55)(2,54)(3,53)(4,52)(5,51)(6,50)(7,49)(8,48)(9,47)(10,46)(11,45)(12,44)(13,43)(14,42)(15,41)(16,40)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(28,56)(57,110)(58,109)(59,108)(60,107)(61,106)(62,105)(63,104)(64,103)(65,102)(66,101)(67,100)(68,99)(69,98)(70,97)(71,96)(72,95)(73,94)(74,93)(75,92)(76,91)(77,90)(78,89)(79,88)(80,87)(81,86)(82,85)(83,112)(84,111), (1,35,8,56,15,49,22,42)(2,50,9,43,16,36,23,29)(3,37,10,30,17,51,24,44)(4,52,11,45,18,38,25,31)(5,39,12,32,19,53,26,46)(6,54,13,47,20,40,27,33)(7,41,14,34,21,55,28,48)(57,104,64,97,71,90,78,111)(58,91,65,112,72,105,79,98)(59,106,66,99,73,92,80,85)(60,93,67,86,74,107,81,100)(61,108,68,101,75,94,82,87)(62,95,69,88,76,109,83,102)(63,110,70,103,77,96,84,89), (1,104,22,97,15,90,8,111)(2,105,23,98,16,91,9,112)(3,106,24,99,17,92,10,85)(4,107,25,100,18,93,11,86)(5,108,26,101,19,94,12,87)(6,109,27,102,20,95,13,88)(7,110,28,103,21,96,14,89)(29,72,50,65,43,58,36,79)(30,73,51,66,44,59,37,80)(31,74,52,67,45,60,38,81)(32,75,53,68,46,61,39,82)(33,76,54,69,47,62,40,83)(34,77,55,70,48,63,41,84)(35,78,56,71,49,64,42,57) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28),(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)], [(1,55),(2,54),(3,53),(4,52),(5,51),(6,50),(7,49),(8,48),(9,47),(10,46),(11,45),(12,44),(13,43),(14,42),(15,41),(16,40),(17,39),(18,38),(19,37),(20,36),(21,35),(22,34),(23,33),(24,32),(25,31),(26,30),(27,29),(28,56),(57,110),(58,109),(59,108),(60,107),(61,106),(62,105),(63,104),(64,103),(65,102),(66,101),(67,100),(68,99),(69,98),(70,97),(71,96),(72,95),(73,94),(74,93),(75,92),(76,91),(77,90),(78,89),(79,88),(80,87),(81,86),(82,85),(83,112),(84,111)], [(1,35,8,56,15,49,22,42),(2,50,9,43,16,36,23,29),(3,37,10,30,17,51,24,44),(4,52,11,45,18,38,25,31),(5,39,12,32,19,53,26,46),(6,54,13,47,20,40,27,33),(7,41,14,34,21,55,28,48),(57,104,64,97,71,90,78,111),(58,91,65,112,72,105,79,98),(59,106,66,99,73,92,80,85),(60,93,67,86,74,107,81,100),(61,108,68,101,75,94,82,87),(62,95,69,88,76,109,83,102),(63,110,70,103,77,96,84,89)], [(1,104,22,97,15,90,8,111),(2,105,23,98,16,91,9,112),(3,106,24,99,17,92,10,85),(4,107,25,100,18,93,11,86),(5,108,26,101,19,94,12,87),(6,109,27,102,20,95,13,88),(7,110,28,103,21,96,14,89),(29,72,50,65,43,58,36,79),(30,73,51,66,44,59,37,80),(31,74,52,67,45,60,38,81),(32,75,53,68,46,61,39,82),(33,76,54,69,47,62,40,83),(34,77,55,70,48,63,41,84),(35,78,56,71,49,64,42,57)]])
49 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 7A | 7B | 7C | 8A | 8B | 14A | 14B | 14C | 14D | 14E | 14F | 14G | ··· | 14L | 28A | ··· | 28F | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 4 | 28 | 28 | 2 | 2 | 28 | 28 | 28 | 28 | 56 | 2 | 2 | 2 | 8 | 8 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
49 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | D14 | D14 | D28 | D4.9D4 | D4×D7 | D28.1D4 |
| kernel | D28.1D4 | D28⋊4C4 | C7×C4.D4 | C8.D14 | C28.17D4 | D4⋊6D14 | Dic14 | D28 | C22×C14 | C4.D4 | M4(2) | C2×D4 | C23 | C7 | C4 | C1 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 3 | 6 | 3 | 12 | 2 | 6 | 3 |
Matrix representation of D28.1D4 ►in GL6(𝔽113)
| 34 | 89 | 0 | 0 | 0 | 0 |
| 59 | 88 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 0 | 0 | 0 |
| 0 | 0 | 98 | 98 | 0 | 0 |
| 0 | 0 | 98 | 0 | 98 | 0 |
| 0 | 0 | 67 | 55 | 79 | 15 |
| 0 | 103 | 0 | 0 | 0 | 0 |
| 79 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 98 | 83 | 0 | 0 |
| 0 | 0 | 15 | 15 | 0 | 0 |
| 0 | 0 | 21 | 74 | 19 | 63 |
| 0 | 0 | 51 | 101 | 75 | 94 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 111 | 0 | 0 |
| 0 | 0 | 8 | 1 | 0 | 0 |
| 0 | 0 | 24 | 20 | 54 | 72 |
| 0 | 0 | 7 | 67 | 68 | 59 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 66 | 75 | 5 | 82 |
| 0 | 0 | 16 | 19 | 39 | 72 |
| 0 | 0 | 24 | 20 | 54 | 72 |
| 0 | 0 | 4 | 74 | 8 | 87 |
G:=sub<GL(6,GF(113))| [34,59,0,0,0,0,89,88,0,0,0,0,0,0,15,98,98,67,0,0,0,98,0,55,0,0,0,0,98,79,0,0,0,0,0,15],[0,79,0,0,0,0,103,0,0,0,0,0,0,0,98,15,21,51,0,0,83,15,74,101,0,0,0,0,19,75,0,0,0,0,63,94],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,8,24,7,0,0,111,1,20,67,0,0,0,0,54,68,0,0,0,0,72,59],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,66,16,24,4,0,0,75,19,20,74,0,0,5,39,54,8,0,0,82,72,72,87] >;
D28.1D4 in GAP, Magma, Sage, TeX
D_{28}._1D_4 % in TeX
G:=Group("D28.1D4"); // GroupNames label
G:=SmallGroup(448,280);
// by ID
G=gap.SmallGroup(448,280);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,254,219,58,1123,570,136,1684,438,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^28=b^2=1,c^4=a^14,d^2=a^21,b*a*b=a^-1,c*a*c^-1=a^15,a*d=d*a,c*b*c^-1=a^21*b,d*b*d^-1=a^7*b,d*c*d^-1=a^7*c^3>;
// generators/relations